Definition and Remark

The category ACmkACm_k of multiplicative kk-groups forma a subcategory of the category ACkAC_k of affine commutative kk-groups which is stable under forming subgroups, quatients, extensions (the set of these properties says that the subcategory is thick) and limits.

ACmkACm_k is (contravariant) equivalent to the category of Galois modules: To GG corresponds the Galois module D^(G⊗kks)(ks)=Grks(G⊗kks,μks)\hat D(G\otimes_k k_s)(k_s)=Gr_{k_s}(G\otimes_k k_s,\mu_{k_s}).

If EE is an étale kk-group, then D(E)D(E) is multiplicative and D^(D(E))=E\hat D(D(E))=E. And we have D(D(E))=ED(D(E))=E. The duality is hence given by E→D(E)E\to D(E) , G→D(G)G\to D(G) without reference to formal groups.